1-Having understood what is meant by
determinism, initial conditions, and uncertainty of measurements, you can now
learn about dynamical instability, which to most physicists is the same in
meaning as chaos.

5- That these mathematical orbit equations
are deterministic means, of course, that by knowing the initial conditions---in
this case, the positions and velocities of the planets at a given starting
time---you find out the positions and speeds of the planets at any time in the
future or past.

6-Of course, it is impossible to actually
measure the initial positions and speeds of the planets to infinite precision,
even using perfect measuring instruments, since it is impossible to record any
measurement to infinite precision. Thus there always exists an imprecision,
however small, in all astronomical predictions made by the equation forms of
Newton's laws.

7- Up until the time of Poincaré, the lack
of infinite precision in astronomical predictions was considered a minor
problem, however, because of a tacit assumption made by almost all physicists
at that time.

8- The assumption was that if you could
shrink the uncertainty in the initial conditions---perhaps by using finer
measuring instruments---then any imprecision in the prediction would shrink in
the same way.

9- In other words, by putting more precise
information into Newton's laws, you got more precise output for any later or
earlier time. Thus it was assumed that it was theoretically possible to obtain
nearly-perfect predictions for the behavior of any physical system.

11- By examining the mathematical equations,
he found that although certain simple astronomical systems did indeed obey the
"shrink-shrink" rule for initial conditions and final predictions, other
systems did not.

12- The astronomical systems which did not
obey the rule typically consisted of three or more astronomical bodies with
interaction between all three. For these types of systems, Poincaré showed that
a very tiny imprecision in the initial conditions would grow in time at an
enormous rate.

14- Poincaré mathematically proved that this
"blowing up" of tiny uncertainties in the initial conditions into enormous
uncertainties in the final predictions remained even if the initial
uncertainties were shrunk to smallest imaginable size.

15- That is, for these systems, even if you
could specify the initial measurements to a hundred times or a million times
the precision, etc., the uncertainty for later or earlier times would not
shrink, but remain huge.

16- The gist of Poincaré's mathematical
analysis was a proof that for these "complex systems," the only way to obtain
predictions with any degree of accuracy at all would entail specifying the
initial conditions to absolutely infinite precision.

17- For these astronomical systems, any
imprecision at all, no matter how small, would result after a short period of
time in an uncertainty in the deterministic prediction which was hardly any
smaller than if the prediction had been made by random chance.

19- Because long-term mathematical
predictions made for chaotic systems are no more accurate that random chance,
the equations of motion can yield only short-term predictions with any degree
of accuracy.

20- Although Poincaré's work was considered
important by some other foresighted physicists of the time, many decades would
pass before the implications of his discoveries were realized by the science
community as a whole. One reason was that much of the community of physicists
was involved in making new discoveries in the new branch of physics called
quantum mechanics, which is physics extended to the atomic realm.